[PPT] - Basic Concepts: Magnetism of electrons J. M. D. Coey School of PowerPoint Presentation

SLIDE 1

Basic Concepts: Magnetism of electrons

J. M. D. Coey

School of Physics and CRANN, Trinity College Dublin Ireland. 1. Spin and orbital moment of the electron 2. Paramagnetism of localized electrons 3. Precession and resonance 4. The free electron gas 5. Pauli paramagnetism 6. Landau diamagnetism

www.tcd.ie/Physics/Magnetism Comments and corrections please: jcoey@tcd.ie

SLIDE 2

This series of three lectures covers basic concepts in magnetism; Firstly magnetic moment, magnetization and the two magnetic fields are presented. Internal and external fields are distinguished. The main characteristics of ferromagnetic materials are briefly

introduced. Magnetic energy and forces are discussed. SI units

are explained, and dimensions are given for magnetic, electrical and other physical properties. Then the electronic origin of paramagnetism of non-interacting electrons is calculated in the localized and delocalized limits. The multi-electron atom is analysed, and the influence of the local crystalline environment on its paramagnetism is explained. Assumed is an elementary knowledge of solid state physics, electromagnetism and quantum mechanics.

SLIDE 3

1. Magnetism of the electron

ESM Cluj 2015

SLIDE 4

Einstein-de Hass Experiment

Demonstrates the relation between magnetism and angular momentum. A ferromagnetic rod is suspended on a torsion fibre. The field in the solenoid is reversed, switching the direction of magnetization of the rod. An angular impulse is delivered due to the reversal

f the angular momentum of the electrons-

conservation of angular momentum. Ni has 28 electrons, moment per Ni is that of 0.6e Three huge paradoxes; — Amperian surface currents 100 years ago — Weiss molecular field — Bohr - van Leeuwen theorem

ESM Cluj 2015

SLIDE 5

The electron

The magnetic properties of solids derive essentially from the magnetism of their electrons. (Nuclei also possess magnetic moments, but they are ≈ 1000 times smaller). An electron is a point particle with: mass me = 9.109 10-31 kg charge

e = -1.602 10-19 C

intrinsic angular momentum (spin) ½ħ = 0.527 10-34 J s On an atomic scale, magnetism is always associated with angular momentum. Charge is negative, hence the angular momentum and magnetic moment are oppositely directed

(a) (b)

← ←

Orbital moment Spin

m l

I

The same magnetic moment, the Bohr Magneton, µB = eħ/2me = 9.27 10-24 Am2 is associated with ½ħ of spin angular momentum or ħ of

rbital angular momentum

ESM Cluj 2015

SLIDE 6

Origin of Magnetism

1930 Solvay conference

At this point it seems that the whole of chemistry and much of physics is understood in principle. The problem is that the equations are much to difficult to solve….. P. A. M. Dirac

ESM Cluj 2015

SLIDE 7

Orbital and Spin Moment

Magnetism in solids is due to the angular momentum of electrons on atoms. Two contributions to the electron moment:

Orbital motion about the nucleus
Spin- the intrinsic (rest frame) angular

m momentum. m m = - (µB /ħ)(l + 2s)

(b)

←

(a)

←

ESM Cluj 2015

SLIDE 8

Orbital moment

Circulating current is I; I = -e/τ = -ev/2πr The moment* is m = IA m = -evr/2 Bohr: orbital angular momentum l is quantized in units of

ħ; h is Planck’s constant = 6.626 10-34 J s; ħ = h/2π = 1.055 10-34 J s. |l| = nħ

Orbital angular momentum: l = mer x v Units: J s Orbital quantum number l, lz= mlħ ml =0,±1,±2,...,±l so mz = -ml(eħ/2me)

The Bohr model provides us with the natural unit of magnetic moment

Bohr magneton µB = (eħ/2me) µB = 9.274 10-24 A m2 mz = mlµB In general m m = γl γ = gyromagnetic ratio Orbital motion γ = -e/2me * Derivation can be generalized to noncircular orbits: m = IA for any planar orbit.

ESM Cluj 2015

SLIDE 9

g-factor; Bohr radius; energy scale

The g-factor is defined as the ratio of magnitude of m in units of µB to magnitude of l in units of ħ. g = 1 for orbital motion The Bohr model also provides us with a natural unit of length, the Bohr radius a0 = 4πε0ħ2/mee2

a0 = 52.92 pm

and a natural unit of energy, the Rydberg R0 R0 = (m/2ħ2)(e2/4πε0)2 R0 = 13.606 eV

ESM Cluj 2015

SLIDE 10

Spin moment

Spin is a relativistic effect. Spin angular momentum s Spin quantum number s s = ½ for electrons Spin magnetic quantum number ms ms = ±½ for electrons sz = msħ ms= ±½ for electrons For spin moments of electrons we have: γ = -e/me g ≈ 2

m = -(e/me)s mz = -(e/me)msħ = ±µB

More accurately, after higher order corrections: g = 2.0023 mz = 1.00116µB

m = - (µB/ħ)(l + 2s)

An electron will usually have both orbital and spin angular momentum

ESM Cluj 2015

SLIDE 11

Quantized mechanics of spin

In quantum mechanics, we represent physical observables by operators – differential or matrix. e.g. momentum p = -iħ∇; energy p2/2me = -ħ2∇2/2me n magnetic basis states ⇒ n x n Hermitian matrix, Aij=A*

ji Spin operator (for s = ½ )

s = σħ/2 Pauli spin matrices

Electron: s = ½ ⇒ ms=±½ i.e spin down and spin up states Represented by column vectors: |↓〉 = |↑〉 =

s |↑〉 = - (ħ/2) |↑〉 ; s|↓〉 = (ħ/2)|↓〉

Eigenvalues of s2: s(s+1)ħ2 The fundamental property of angular momentum in QM is that the operators satisfy the commutation relations:

r

Where [A,B] = AB - BA and [A,B] = 0 ⇒ A and B’s eigenvalues can be measured simultaneously

[s2,sz] = 0

ESM Cluj 2015

SLIDE 12

Quantized spin angular momentum of the electron

1/2

1/2

ms

S

z g√[s(s+1)]ħ2

H

1/2 1/2

s = ½

2µ0µBH
ħ/2

ħ/2

The electrons have only two eigenstates, ‘spin up’(↑, ms = -1/2) and ‘spin down’ (↓, ms = 1/2), which correspond to two possible orientations of the spin moment relative to the applied field.

ESM Cluj 2015

SLIDE 13

2. Paramagnetism of localized electrons

ESM Cluj 2015

SLIDE 14

Populations of the energy levels are given by Boltzmann statistics; ∝ exp{-Ei/kΒT}. The thermodynamic average 〈m〉 is evaluated from these Boltzmann populations. 〈m 〉 = [µBexp(x) - µBexp(-x)] [exp(x) + exp(-x)] where x = µ0µBH/kBT. 〈m 〉 = µBtanh(x) Note that to approach saturation x ≈ 2 At T = 300 K, µ0H. = 900 T At T = 1K , µ0H. = 3 K.

Useful conversion 1 TµB = 0.672 (µB/kB)

Spin magnetization of localized electrons

2 4 6 8 0.2 0.4 0.6 ∞ 0.8 1/2 1.0 2

,

x

Slope 1

ESM Cluj 2015

SLIDE 15

In small fields, tanh(x) ≈ x, hence the susceptibility χ = N〈m 〉/H (N is no of electrons m-3) χ = µ0NµB

2/kBT

This is the famous Curie law for susceptibility, which varies as T-1. In other terms χ = C/T, where C = µ0NµB

2/kB

is a constant with dimensions of temperature; Assuming an electron density N of 6 1028 m-3 gives a Curie constant C ≈ 0.5 K. The Curie law susceptibility at room temperature is of order 10-3.

Curie-law susceptibility of localized electrons

T 1/χ

Slope C

ESM Cluj 2015

SLIDE 16

3. Spin precession and resonance

ESM Cluj 2015

SLIDE 17

Electrons in a field; paramagnetic resonance

1/2 1/2 ms S

s = ½

gµ0µBH

At room temperature there is a very slight difference in thermal populations of the two spin states (hence the very small spin susceptibility of 10-3). The relative population difference is x = gµ0µBH/2kBT At resonance, energy is absorbed from the rf field until the populations are equalized. The resonance condition is hf = gµ0µBH f/µ0H = gµB/h [= geħ/2meh = e/2πme] Spin resonance frequency is 28 GHz T-1 hf

ESM Cluj 2015

SLIDE 18

m = γl [γ = -e/me] Γ = m x B Γ = dl/dt (Newton’s law) dm /dt = γ m x B = γ ex ey ez

mx my mz 0 0 Bz

Solution is m(t) = m ( sinθ cosωLt, sinθ sinωLt, cosθ ) where ωL = γBz Magnetic moment precesses at the Larmor precession frequency fL = γB/2π dM/dt = γM x B – αeM x dM/dt 28 GHz T-1 for spin Γ = m x B

m m

BZ dl/dt dmx/dt = γmyBz dmy/dt = -γmxBz dmz/dt = 0 θ

Electrons in a field - Larmor precession

ESM Cluj 2015

SLIDE 19

Free electrons follow cyclotron orbits in a magnetic field. Electron has velocity v then it experiences a Lorentz force F = -ev × B The electron executes circular motion about the direction of B (tracing a helical path if v|| ≠ 0) Cyclotron frequency fc = v⊥/2πr fc = eB/2πme Electrons in cyclotron orbits radiate at the cyclotron frequency Example: — Microwave oven Since γe = -(e/me), the cyclotron and Larmor and epr frequencies are all the same for electrons; 28.0 GHz T-1

Electrons in a field – Cyclotron resonance

ESM Cluj 2015

SLIDE 20

4. The free electron gas

ESM Cluj 2015

SLIDE 21

ESM Cluj 2015

We apply quantum mechanics to the electrons. They have spin ½ , and thus there are two magnetic states, ms = ½ (spin up ↑) and ms = - ½ (spin down ↓), for every electron. Suppose the electrons are confined in a box of volume V, where the potential is constant, U0 Electrons are represented by a wavefunction ψ(r) where ψ*(r)ψ(r)dV is the probability of finding an electron in a volume dV. Schrödinger’s equation Hψ(r) = E ψ(r) {p2/2m + U0}ψ(r) = E ψ(r) but p → -i∇ {- 2∇2/2m + U0}ψ(r) = E ψ(r) Solutions are ψk(r) = (1/V1/2) exp ik.r

Normalization wave vector

The wave vector of the electron k = 2π/λ Its momentum; -i∇ψ(r) = kψ(r) , is k.

Free electron model

SLIDE 22

Only certain values of k are allowed. The boundary condition is that L is an integral number of wavelengths. ki = 0, ±2π/L, ±4π/L, ±6π/L …….. The allowed states are represented by points in k-space There is just one state in each volume (2π/L)3 of k-space, And at most two electrons, one spin up ↑ and one spin down ↓, can occupy each state. Electrons are fermions. The energy of an electron in the box is E = p2/2me Ek = (k)2/2me + U0 L E - U0 k

Free -electron parabola

Free electron model

ESM Cluj 2015

SLIDE 23

The points in k-space are very closely spaced; There are N ~ 1022 electrons in a macroscopic sample, so k is effectively a continuous variable. At temperature T = 0, we fill up all the lowest energy states, with two electrons per state, up to the Fermi level. The energy of the last electron is the Fermi energy EF. The wavelength of the last electron is the Fermi wavelength kF. The N occupied states are contained within the Fermi

surface. In the free-electron model this surface is a sphere.

kx

● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●

ky We calculate EF. N = (4π/3)kF

3 x 2/(2π/L)3 → kF = (3π2N/V)1/3

(EF - U0) = (kF)2/2m = (2/2m) (3π2n)2/3 where n = N/V For Cu, (EF - U0) ≈ 7 eV. TF is defined by kTF = EF. For Cu, TF ≈ 80,000 K (1 eV =11605 K) The Fermi velocity vF = kF/m For Cu, vF ≈ 1.6 106 m s-1

Free electron model

ESM Cluj 2015

SLIDE 24

A useful concept is the density of states, the number of states per unit sample volume, as a function of k or E. The number between k and k + δk is D ( k ) δ k = 4 π k 2 δ k ( L / 2 π ) 3 x 2 Now E = 2k2/2m → δE/δk = 2k/m The number between E and E + δE is D(E)δE = 4πk2(L/2π)3 x 2/(2k/m) δE D(E)δE = (Vm/π22)(2mE/2)1/2δE E At the Fermi level * D(EF) = (3/2)n/EF Units of D(EF) are states J- m-3 ( or states eV-1 m-3) State occupancy when T > 0 is given by the Fermi function f(E) = 1/[exp(E - µ)/kBT + 1] (5) The chemical potential µ is fixed by ∫0

∞ f(E)dE = 1

Note: µ = EF at T = 0; also j = (σ/e)∇µ D(E) E f(E) kBT

Free electron model

ESM Cluj 2015

SLIDE 25

ESM Cluj 2015

Some physical properties can be explained solely in terms of the density of states at the Fermi level D(EF) Only electrons within ~ kBT of the Fermi level can be thermally excited. The number of these electrons is D(EF) kBT The increase in energy U(T) - U(0) is ~ D(EF) (kBT)2 Cel = dU/dT ≈ 2D(EF) kB

2T

The exact result is Cel = (π2/3)D(EF) kB

2T = γT

When T << ΘD (the Debye temperature) C = γT + βT3 Note that the electronic entropy Sel = ∫0

T (Cel/T’) dT’ [recall δQ = TδS]

According to the third law of thermodynamics, S→ 0 as T→ 0 .

Electronic contribution

Lattice contribution

EF E D(E)

kBT

Electronic specific heat

SLIDE 26

D↑,↓(E) E E ↓ ↑ E E ↓ ↑ H

±µ0µBH

EF The splitting is really very small, ~ 10-5 of the bandwidth in a field of 1 T. M = µB(N↑ - N↓)/V Note M is magnetic moment per unit volume At T = 0, the change in population in each band is ΔN = ½ D(EF)µ0µBH M = 2µB ΔN = D(EF)µ0µB

2H The dimensionless susceptibility χ = M/H

χPauli = D(EF)µ0µB

2 It is ~ 10-5 and independent of T

Pauli susceptibility

We now show the ↑ and ↓ density of states separately. They split in a field B = µ0H

ESM Cluj 2015

SLIDE 27

Landau diamagnetism

Free electron model was used by Landau to calculate the orbital diamagnetism of conduction electrons. The result is: exactly one third of the Pauli susceptibility, and opposite in sign. The real band structure is taken into account in an approximate way by renormalizing the electron mass. Replace me by an effective mass m* Then χL = -(1/3)(me/m) χP In some semimetals such as graphite or bismuth, m can be ≈ 0.01 me, hence the diamagnetism of the conduction electrons may sometimes be the dominant contribution to the susceptibility. (χL = -4 10-4 for graphite) In the free-electron model, D(EF) = (3/2)n/EF Hence χPauli = {3nµ0µB

2/2EF }[1 + cT2 + ….] (Compare Curie law nµ0µB 2/kBT)

The ratio of electronic specific heat coefficient to Pauli susceptibility in the nearly-free, independent electron approximation should be a constant R.

ESM Cluj 2015

SLIDE 28

Landau diamagnetism

Curie Pauli Landau

ESM Cluj 2015

SLIDE 29

Density of states in other dimension

ESM Cluj 2015

D(ε) ∝ ε1/2 D(ε) = constant D(ε) ∝ ε-1/2 Discreet levels 3-d solid

SLIDE 30

Let B = Bz, A = (0, xB, 0), V(r) = 0 and m = m* Canonical momentum p = p - qA Schrodinger’s equation ωc = eB/m*, x0 = -ħky/eB E’ = E - (ħ2/2me)kz

2

The motion is a plane wave along Oz, plus a simple harmonic oscillation at fc = ωc/2, in the plane, where ωc = eB/me

Quantum oscillations

ESM Cluj 2015

1 2m∗ p2

x + (py + exB)2 + p2 z

ψ = εψ,

SLIDE 31

When a magnetic field is applied, the states in the Fermi sphere collapse onto a series

f tubes. Each tube corresponds to one Landau level (n - value). As the field increases,

the tubes expand and the outer one empties periodically as field increases. An oscillatory variation in 1/B2 of magnetization (de Haas - van Alphen effect) or of conductivity (Shubnikov - de Haas effect) appears. From the period, it is possible to deduce the cross section area of the Fermi surface normal to the tubes.

De Haas van Alfen effect

ESM Cluj 2015

SLIDE 32

ESM Cluj 2015

SLIDE 33

Maxwell’s equations relate magnetic and electric fields to their sources. The other fundamental relation of electrodynamics is the expression for the force on a moving particle with charge q, F = q(E + v∧B) The two terms are respectively the Coulomb and Lorentz forces. The latter gives the torque equation Γ = m∧B The corresponding Hamiltonian for the particle in a vector potential A representing the magnetic field B (B = ∇∧Α) and a scalar potential φε representing the electric field E (E = -∇φe) is H = (1/2m)(p - qA)2 +qφe

Theory of electronic magnetism

ESM Cluj 2015

SLIDE 34

The Hamiltonian of an electron with electrostatic potential energy V(r) = -eφe is H = (1/2m)(p + eA)2 +V(r) Now (p + eA)2 = p2 + e2A2 + 2eA.p since A and p commute when ∇.A = 0. So H = [p2/2m +V(r)] + (e/m)A.p + (e2/2m)A2 H = H 0 + H 1 + H 2 where H0 is the unperturbed Hamiltonian, H1 gives the paramagnetic response of the orbital moment and H2 describes the small diamagnetic response. Consider a uniform field B along z. Then the vector potential in component form is A = (1/2) (-By, Bx, 0), so B = ∇∧Α = = ez(∂Ay/∂x - ∂Ax/∂y) = ezB. More generally A = (1/2)B∧r Now (e/m)A.p = (e/2m)B∧r.p = (e/2m)B.r∧p = (e/2m)B.l since l = r∧p. The second terms in the Hamiltonian is then the Zeeman interaction for the orbital moment H 1 = (µB/)B.l The third term is (e2/8m) (B∧r)2 = (e2/8m2)B2(x2+y2). If the orbital is spherically symmetric, <x2>= <y2>= <r2>/3.The corresponding energy E = (e2B2/12m) <r2>. Since M = -∂E/∂B and susceptibility χ = µ0NM/B, It follows that the orbital diamagnetic susceptibility is χ = µ0Ne2 <r2>/6m.

Orbital moment

ESM Cluj 2015

SLIDE 35

Spin moment

The time-dependent Schrödinger equation

(2/2m)∇2ψ + Vψ = i∂ψ/∂t

is not relativistically invariant because the operators ∂/∂t and ∂/∂x do not appear to the same power. We need to use a 4-vector X = (ct, x, y, z) with derivatives ∂/∂X. Dirac discovered the relativistic quantum mechanical theory of the electron, which involves the Pauli spin

perators σI, with coupled equations for electrons and positrons. The nonrelativistic limit of the theory,

including the interaction with a magnetic field B represented by a vector potential A can be written as H = [(1/2m)(p + eA)2 +V(r)] - p4/8m3c2 + (e/m)B.s + (1/2m2c2r)(dV/dr) - (1/4m2c2)(dV/dr) ∂/∂r

The second term is a higher-order correction to the kinetic energy
The third term is the interaction of the electron spin with the magnetic field, so that the complete

expression for the Zeeman interaction of the electron is H Z = (µB/)B.(l + 2s)

ESM Cluj 2015

SLIDE 36

The factor 2 is not quite exact. The expression is 2(1 + α/2π - .....) ≈ 2.0023, where α = e2/4πε0hc≈ 1/137 is the fine-structure constant.

The fourth term is the spin-orbit ineteraction., which for a central potential V(r) = -Ze2/4πε0r with Ze as

the nuclear charge becomes -Ze2µ0l.s/8πm2r3 since µ0ε0 = 1/c2. In an atom <1/r3> ≈ (0.1 nm)3 so the magnitude of the spin-orbit coupling λ is 2.5 K for hydrogen (Z = 1), 60 K for 3d elements (Z ≈ 25), and 160 K for actinides (Z ≈ 65). In a noncentral potential, the spin-orbit interaction is (s∧∇V).p

The final term just shifts the levels when l = 0

Spin moment

ESM Cluj 2015

SLIDE 37

Magnetism and relativity

The classification of interactions according to their relativistic character is based on the kinetic energy E = mc2√[1 + (v2/c2)] The order of magnitude of the velocity of electrons in solids is αc. Expanding the equation in powers of c E = mc2 + (1/2)α2mc2 - (1/8)α4mc2 Here the rest mass of the electron, mc2 = 511 keV; the second and third terms, which represent the order of magnitude of electrostatic and magnetostatic energies are respectively 13.6 eV and 0.18 meV. Magnetic dipolar interactions are therefore of

rder 2 K. (1 eV = 11605 K)

ESM Cluj 2015

SLIDE 38

4 Be

9.01 2 + 2s0

12Mg

24.21 2 + 3s0

2 He

4.00

10Ne

20.18

24Cr

52.00

3 + 3d3 312

19K

38.21

1 + 4s0

11Na

22.99 1 + 3s0

3 Li

6.94 1 + 2s0

37Rb

85.47 1 + 5s0

55Cs

132.9 1 + 6s0

38Sr

87.62

2 + 5s0

56Ba

137.3

2 + 6s0

59Pr

140.9 3 + 4f2

1 H

1.00

5 B

10.81

9 F

19.00

17Cl

35.45

35Br

79.90

21Sc

44.96

3 + 3d0

22Ti

47.88

4 + 3d0

23V

50.94

3 + 3d2

26Fe

55.85

3 + 3d5

1043

27Co

58.93

2 + 3d7

1390

28Ni

58.69

2 + 3d8

629

29Cu

63.55

2 + 3d9

30Zn

65.39

2 + 3d10

31Ga

69.72

3 + 3d10

14Si

28.09

32Ge

72.61

33As

74.92

34Se

78.96

6 C

12.01

7 N

14.01

15P

30.97

16S

32.07

18Ar

39.95

39Y

88.91

2 + 4d0

40Zr

91.22

4 + 4d0

41Nb

92.91

5 + 4d0

42Mo

95.94

5 + 4d1

43Tc

97.9

44Ru

101.1

3 + 4d5

45Rh

102.4

3 + 4d6

46Pd

106.4

2 + 4d8

47Ag

107.9

1 + 4d10

48Cd

112.4

2 + 4d10

49In

114.8

3 + 4d10

50Sn

118.7

4 + 4d10

51Sb

121.8

52Te

127.6

53I

126.9

57La

138.9

3 + 4f0

72Hf

178.5

4 + 5d0

73Ta

180.9

5 + 5d0

74W

183.8

6 + 5d0

75Re

186.2

4 + 5d3

76Os

190.2

3 + 5d5

77Ir

192.2

4 + 5d5

78Pt

195.1

2 + 5d8

79Au

197.0

1 + 5d10

61Pm

145

70Yb

173.0 3 + 4f13

71Lu

175.0 3 + 4f14

90Th

232.0 4 + 5f0

91Pa

231.0 5 + 5f0

92U

238.0 4 + 5f2

87Fr

223

88Ra

226.0

2 + 7s0

89Ac

227.0

3 + 5f0

62Sm

150.4 3 + 4f5

105

66Dy

162.5 3 + 4f9 179 85

67Ho

164.9 3 + 4f10 132 20

68Er

167.3 3 + 4f11 85 20

58Ce

140.1 4 + 4f0

13

Ferromagnet TC > 290K Antiferromagnet with TN > 290K 8 O

16.00 35

65Tb

158.9 3 + 4f8 229 221

64Gd

157.3 3 + 4f7 292

63Eu

152.0 2 + 4f7 90

60Nd

144.2 3 + 4f3 19

66Dy

162.5 3 + 4f9 179 85

Atomic symbol Atomic Number Typical ionic change Atomic weight Antiferromagnetic TN(K) Ferromagnetic TC(K) Antiferromagnet/Ferromagnet with TN/TC < 290 K Metal Radioactive

Magnetic Periodic Table

80Hg

200.6

2 + 5d10

93Np

238.0 5 + 5f2

94Pu

244

95Am

243

96Cm

247

97Bk

247

98Cf

251

99Es

252

100Fm

257

101Md

258

102No

259

103Lr

260

36Kr

83.80

54Xe

83.80

81Tl

204.4

3 + 5d10

82Pb

207.2

4 + 5d10

83Bi

209.0

84Po

209

85At

210

86Rn

222

Nonmetal Diamagnet Paramagnet BOLD Magnetic atom 25Mn

55.85

2 + 3d5

96

20Ca

40.08

2 + 4s0

13Al

26.98

3 + 2p6

69Tm

168.9 3 + 4f12 56

ESM Cluj 2015